This exposition is primarily a survey of the elementary yet subtle innovations of several mathematicians between 1929 and 1934 that led to partial and then complete solutions to Hilbert's Seventh Problem (from the International Congress of Mathematicians in Paris, 1900). This volume is suitable for both mathematics students wishing to experience how different mathematical ideas can come together to establish results and for research mathematicians interested in the fascinating progression of mathematical ideas that solved Hilbert's problem and established a modern theory of transcendental numbers.